Question

True or False \(\sin 182^{\circ}=\cos 2^{\circ}\)

Short answer

False

Step-by-step solution

Understand the Given Problem

To determine whether the statement \(\sin 182^{\circ} = \cos 2^{\circ}\) is true or false, first, recall the relationship between sine and cosine in terms of complementary angles.

Recall Trigonometric Identity

Use the identity \(\sin(90^{\circ} + x) = \cos(x)\).

Apply the Identity to the Sine Term

Set \(x = 92^{\circ}\). Then \(\sin(182^{\circ}) = \sin(90^{\circ} + 92^{\circ})\).

Simplify Using the Identity

By applying the identity, \(\sin(90^{\circ} + 92^{\circ}) = \cos(92^{\circ})\).

Compare the Results

Now, compare \(\cos(92^{\circ})\) with \(\cos(2^{\circ})\). These two are not equal.

Conclusion

Since \(\cos(92^{\circ})\) is not equal to \(\cos(2^{\circ})\), the statement \(\sin 182^{\circ} = \cos 2^{\circ}\) is false.

Key concepts

Sine and Cosine Relationship

In trigonometry, the sine and cosine functions have a special relationship that is useful for solving many problems. One key identity is that \(\sin(90^{\circ} - x) = \cos(x)\). This means if you know the value of either sine or cosine for a given angle, you can easily find the other by using this complementary angle relationship.
For example, if \(\cos(30^{\circ}) = \sin(60^{\circ})\), this shows how angles add up to 90 degrees.
Understanding this relationship helps in breaking down more complex trigonometric expressions into simpler forms.

Complementary Angles

Complementary angles are two angles that add up to 90 degrees. In terms of trigonometry, these complementary angles help us find equivalent values of sine and cosine. For instance, if \(\theta\) is an angle, then \(\sin(90^{\circ} - \theta )\) is the same as \(\cos(\theta )\).
This identity comes in handy often; for example, in the exercise you are working on, you can use it to determine that \(\sin(182^{\circ}) \) is equivalent to \(\cos(92^{\circ})\).
Recognizing complementary angles helps in rearranging and simplifying expressions to make calculations easier.

Trigonometric Simplification

Trigonometric simplification is useful in transforming complex expressions into simpler ones. By using identities like the sine-cosine relationship, we can make challenging problems more manageable. For example, if you know \(\sin(90^{\circ} + x) = \cos(x)\), finding the sine or cosine of certain angles becomes easier.
In the given exercise, simplifying \(\sin 182^{\circ}\) using this identity shows that it's equivalent to \(\cos 92^{\circ}\).
This reveals that \(\sin 182^{\circ} \eq \cos 2^{\circ}\). Hence, the statement \sin 182^{\circ} = \cos 2^{\circ}\ is false.
Simplifying expressions helps in verifying and disproving trigonometric equations efficiently.